Problem 76 - College Algebra with Modeling and Visualization

Question

Multiply the polynomials. $$(3+z)(6-4 z)\left(4+2 z^{2}\right)$$

Step-by-Step Solution

Answer

The multiplied result is \(-8z^4 - 12z^3 + 20z^2 - 24z + 72\).

Step 1: Multiply the First Two Terms

First, multiply the polynomials \((3+z)\) and \((6-4z)\). To do this, apply the distributive property or FOIL method:\[(3+z)(6-4z) = 3 \cdot 6 + 3 \cdot (-4z) + z \cdot 6 + z \cdot (-4z)\]This results in:\[18 - 12z + 6z - 4z^2\]Simplifying the expression, we combine like terms:\[18 - 6z - 4z^2\]

Step 2: Multiply the Result with the Third Term

Next, multiply the result from Step 1, \(18 - 6z - 4z^2\), by the third polynomial \((4 + 2z^2)\). Distribute each term in the polynomial:\[(18 - 6z - 4z^2)(4 + 2z^2) = 18 \cdot 4 + 18 \cdot 2z^2 - 6z \cdot 4 - 6z \cdot 2z^2 - 4z^2 \cdot 4 - 4z^2 \cdot 2z^2\]This results in:\[72 + 36z^2 - 24z - 12z^3 - 16z^2 - 8z^4\]

Step 3: Simplify the Expression

Combine like terms in the polynomial obtained from Step 2:- For \(z^4\) term: \(-8z^4\)- For \(z^3\) term: \(-12z^3\)- For \(z^2\) terms: \(36z^2 - 16z^2 = 20z^2\)- For \(z\) term: \(-24z\)- Constant term: \(72\)Thus, the simplified polynomial is:\[-8z^4 - 12z^3 + 20z^2 - 24z + 72\]

Key Concepts

Distributive Property

When multiplying polynomials, one essential tool is the distributive property. It allows us to multiply each term in one polynomial by each term in another. This property is especially useful when dealing with expressions like \[(3+z)(6-4z)\].In this case, we multiply each term in the first polynomial by each term in the second:

• Multiply the constant term 3 by each term in the second polynomial, resulting in the terms \(3 \times 6\) and \(3 \times -4z\).
• Then, multiply the term \(z\) by each term in the second polynomial, yielding \(z \times 6\) and \(z \times -4z\).

This expansion results in four new terms: \(18 - 12z + 6z - 4z^2\), which simplifies to \(18 - 6z - 4z^2\) after combining like terms. The distributive property makes it manageable to break down the multiplication into smaller, simpler steps.

FOIL Method

The FOIL method is a specific application of the distributive property, designed for multiplying two binomials. It's handy because it gives us a straightforward sequence to follow: First, Outer, Inner, Last.In \((3+z)(6-4z)\),we apply FOIL:

• **First:** Multiply the first terms in each binomial, \(3 \times 6\).
• **Outer:** Multiply the outer terms, \(3 \times -4z\).
• **Inner:** Multiply the inner terms, \(z \times 6\).
• **Last:** Multiply the last terms, \(z \times -4z\).

This step-by-step method results in the intermediate expression \(18 - 12z + 6z - 4z^2\), ensuring that all combinations are accounted for in an organized manner. FOIL is a simple strategy to ensure none of the necessary calculations are overlooked.

Combining Like Terms

Once you have multiplied the polynomials using the distributive property or FOIL method, the next important step is combining like terms. This process helps simplify the expression, making it easier to understand and solve.In the expression \[18 - 12z + 6z - 4z^2\], you can combine terms that have the same power of \(z\):

• Terms \(-12z\) and \(+6z\) are like terms because they both have the single power of \(z\). Adding these terms gives \(-6z\).
• The constant \(18\) and \(-4z^2\) have no like terms in this initial grouping, so they remain as they are.

This results in the simplified polynomial: \[18 - 6z - 4z^2\]. Combining like terms is essential in polynomial multiplication, as it tidies up expressions and reduces them to their simplest form, making them more manageable for further calculations or interpretations.